Continuity mistake: A montage scene of newspaper headlines early on has March 12, 1939 as the date. Much later in the film, Jude Law asks Gwyneth Paltrow the date so he can calculate the position of a star. She responds "March 2" but there is absolutely nothing to indicate that nearly a year has passed.
Continuity mistake: When the robots invade New York City and are advancing on Polly in the street, the camera pans down to the robots' stomping feet. There is nothing there but open space for them to tromp on. The camera pans back to Polly taking a picture, then back to the robots' feet. Suddenly, a car has magically appeared in the path of the stomping robots and gets kicked towards her.
Continuity mistake: When Polly stands up while Joe is attacking the robots, she is on the corner of a sidewalk. A few shots later, when Joe flies over her and she says "Joe", she is at a different place on the sidewalk. She was walking really slowly, so it's impossible that she moved that fast.
Add timeDr Wilson
Continuity mistake: When Sky Captain and Polly go visit Dr. Jennings, it is pouring down rain, but when they enter his basement laboratory both of them are completely dry. Granted, they were only exposed to the rain for a few seconds in the distance from the car to the building, but they would still have gotten least a little wet.